Let X and Y be nonempty sets and let h : X × Y †’ R have bounded range in R. Let F : X †’ R and G : Y †’ R be defined by
F(x) := sup{h(x; y) : y ˆˆ Y}; G(y) := sup{h(x; y) : x ˆˆ X}:
Establish the Principle of the Iterated Suprema:
Sup{h(x; y) : x ˆˆ X; y ˆˆ Y} = sup{F(x) : x ˆˆ X} = sup{G(y) : y ˆˆ Y}
We sometimes express this in symbols by

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sup h(x, y) sup sup h(x, y) sup sup h (x, y). x.y